It turns out that the option price calculated using the Black–Scholes–Merton (BSM) model differs from the price formed in the real market. The reason is that the market perceives the volatility of the underlying asset as a function of the strike price and the option’s time to maturity.
The graph shows the so-called volatility smile for currency options.
Implied Volatility is the level of volatility for which the option price calculated by the BSM model equals the market price. In other words, based on observed market option prices, we infer the market’s expectations about volatility. (Here, volatility cannot be calculated using statistical measures of option price fluctuations. This volatility corresponds to a risk-neutral world, which is the framework underlying the BSM formula.)
From the graph, we see that volatility increases as the strike price moves further away from the market (spot) price — moneyness is the key factor here.
Consider a deep out-of-the-money call option. The strike price is high, so demand for such an option exists only if volatility expectations are high.
Similarly, for a deep out-of-the-money put option, the low strike price also reflects expectations of high volatility.
All of this implies that the price movement of the underlying asset does not follow a lognormal distribution, as assumed by the BSM model. In practice, this deviation is observed for currency options.
As noted above, deviations in the tails from the lognormal distribution indicate higher volatility on the right tail for calls and on the left tail for puts.
If market expectations matched a lognormal distribution, the BSM formula would work perfectly and instead of a “volatility smile” we would observe a horizontal line. (For currency options, BSM relies on two assumptions:
- volatility is constant, and
- the asset price evolves smoothly.
These assumptions do not hold in reality, which is why the smile appears.)
It should also be noted that put–call parity implies that the volatility smile is identical for both puts and calls. It can be proven that the deviation between market prices and BSM prices for European puts is the same as for European calls.
Volatility smile for equity options
For equity-based options, the smile curve is skewed.
Two main reasons are cited:
- When a stock price falls (moves away from the strike), financial leverage increases, which raises volatility.
- Crashophobia — after the 1987 market crash, investors became more fearful, leading to a steeper skew.
Accordingly, there is a mismatch between the lognormal distribution and market expectations for equity price movements.
This means that a call option with a higher strike price (K₂) has a lower price than predicted by the BSM model (lower volatility expectations imply a lower option price).
Some important nuances
Moneyness
The volatility smile can be represented in different ways because the strike price K is large or small only relative to the current price.
For currency options, the lowest point of the smile corresponds to the current exchange rate. As the exchange rate changes, the smile shifts left or right.
Therefore, the diagram can also be expressed as a function of K/S₀. Traders often use futures prices, expressing moneyness as K/F₀, to determine whether an option is in or out of the money.
K vs K/S₀
Volatility surfaces
Traders’ belief in mean reversion of volatility is reflected in option market prices. That is, volatility is also a function of the option’s time to maturity.
- If current volatility is below its historical average, traders expect it to increase in the long run. As a result, implied volatility rises with maturity.
- Conversely, if current volatility is high, traders expect conditions to stabilize. This leads to lower implied volatility for longer maturities.
In practice, traders continuously construct a volatility surface from market option prices and trade based on it. The volatility surface is a two-dimensional data table:
The Greeks
Because of the mismatch between market prices and the BSM model, the Greeks (sensitivities) change significantly, making them more difficult to calculate accurately.
Single Large Jump
When the market expects a significant event (for example, the release of drug trial results) that could substantially increase or decrease a stock’s price, the volatility smile becomes bimodal. The expected price distribution becomes discrete (good news or bad news).
In such cases, the implied volatility calculated using BSM differs from true market sentiment because BSM forces option prices to fit a single continuous volatility process.
As a result, indices like VIX (which are calculated using BSM-type frameworks) can show very high volatility when expectations of event risk raise option prices. In this sense, VIX can exaggerate market sentiment about volatility.
Source:
Options, Futures & Other Derivatives — John C. Hull